BIOINORGANIC CHEMISTRY A Short Course Second Edition

92 INSTRUMENTAL METHODS

dynamic and static disorder. Because of protein crystal disorder, the diffraction
pattern fades away at some diffraction angle θmax. The corresponding lattice
distancedmin is determined by Bragg ’ s law as shown in equation 3.9 :

dmin

sin max

=

λ

2 θ

(3.9)

The end result of an X - ray structural determination reports the electron
density in the crystal. The fundamental equation for its calculation follows in
equation 3.10 :

ρπα(xyz) ( ) exp[ ( ) ( )

V

F hkl i hx ky lz i hkl

h k l

=−+++∑∑∑

1

(^2) (3.10)
The electron density at every position x , y , z in the unit cell, ρ ( x y z ), is the
Fourier transform of the structure factor F ( h k l ), which is in turn a function
of the electron density distribution in the unit cell and the integrated intensity
of the refl ected beam, called I ( h k l ). Values of I ( h k l ), the integrated refl ected
beam intensity, are obtained from the diffraction pattern after the application
of correction factors. The term | F ( h k l )| is the structure factor amplitude of
refl ection ( h k l ) including the temperature factor, α ( h k l ) is the phase angle,
andx , y , and z are coordinates in the unit cell. The factor i is the mathematical
imaginary term. The summation occurs over all the discrete directions in which
diffraction by the crystal occurs. Sophisticated computer programs have been
designed to calculate the electron density in the above equation except for the
phase anglesα ( h k l ), which cannot be derived in a straightforward manner
from the diffraction pattern. The “ phase angle ” problem arises because the
X - ray diffraction pattern registers only the intensity of the waves. To work
back to a molecular structure, one needs to know the relative timing when
each wave hits. This is the “ phase data ” — the position of wave crests and
troughs relative to each other. Several methods have been developed to solve
the phase angle problem as listed below.

1. The isomorphous replacement method requires attachment of heavy
   atoms to protein molecules in the crystal. In this method, atoms of high
   atomic number are attached to the protein, and the coordinates of these
   heavy atoms in the unit cell are determined. The X - ray diffraction pattern
   of both the native protein and its heavy atom derivative(s) are deter-
   mined. Application of the so - called Patterson function determines the
   heavy atom coordinates. Following the refi nement of heavy atom param-
   eters, the calculation of protein phase angles proceeds. In the fi nal step
   the electron density of the protein is calculated.


2. The multiple - wavelength anomalous diffraction method (MAD) relies
   upon suffi ciently strong anomalously scattering atoms in the protein
   structure itself. In this method, diffraction data must be collected at a
   number of different wavelengths, usually requiring data collection with